The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation. Find the smallest possible number of independent deputies after the vacation.

In the triangle $\bigtriangleup ABC$ let $D$ be the foot of the altitude from $A$ to the side $BC$ and $I$, $I_A$, $I_C$ be the incenter, $A$-excenter, and $C$-excenter, respectively. Denote by $P\neq B$ and $Q\neq D$ the other intersection points of the circle $\bigtriangleup BDI_C$ with the lines $BI$ and $DI_A$, respectively. Prove that $AP=AQ$. [i]Proposed Michal Jan'ik - Czech Republic[/i]

Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.

Rectangle $ABCD$ has sides in the ratio of $\sqrt2$ to $1$. If $DEC$ is an isosceles right triangle, with $E$ inside the rectangle, find angle $\angle AEB$.

Call a positive integer $n$ [i]tubular [/i] if for any two distinct primes $p$ and $q$ dividing $n, (p + q) | n$. Find the number of tubular numbers less than $100,000$. (Integer powers of primes, including $1, 3$, and $16$, are not considered [i]tubular[/i].)

Prove that for all $n \in \mathbb{Z^+}$ and for all positive real numbers satisfying $a_1a_2...a_n=1$ \[ \displaystyle\sum_{i=1}^{n} \frac{a_i}{\sqrt{{a_i}^4+3}} \leq \frac{1}{2}\displaystyle\sum_{i=1}^{n} \frac{1}{a_i} \]

Let $\mathbb{R}$ be the set of all real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying for all $x,y\in\mathbb{R}$ the equation $f(x)+f(y)=f(f(x)f(y))$.

Let $ABCD$ be a convex quadrilateral, and let $N$ be the midpoint of $BC$. Suppose further that $\angle AND=135^{\circ}$. Prove that $|AB|+|CD|+\frac{1}{\sqrt{2}}\cdot |BC|\ge |AD|.$

[u]Townspeople and Goons[/u] In the city of Lincoln, there is an empty jail, at least two townspeople and at least one goon. A game proceeds over several days, starting with morning. $\bullet$ Each morning, one randomly selected unjailed person is placed in jail. If at this point all goons are jailed, and at least one townsperson remains, then the townspeople win. If at this point all townspeople are jailed and at least one goon remains, then the goons win. $\bullet$ Each evening, if there is at least one goon and at least one townsperson not in jail, then one randomly selected townsperson is jailed. If at this point there are at least as many goons remaining as townspeople remaining, then the goons win. The game ends immediately after any group wins. [b]p1. [/b]Find the probability that the townspeople win if there are initially two townspeople and one goon. [b]p2.[/b] Find the smallest positive integer $n$ such that, if there are initially $2n$ townspeople and $1$ goon, then the probability the townspeople win is greater than $50\%$. [b]p3.[/b] Find the smallest positive integer $n$ such that, if there are initially $n + 1$ townspeople and $n$ goons, then the probability the townspeople win is less than $1\%$. [b]p4[/b]. Suppose there are initially $1001$ townspeople and two goons. What is the probability that, when the game ends, there are exactly $1000$ people in jail? [b]p5.[/b] Suppose that there are initially eight townspeople and one goon. One of the eight townspeople is named Jester. If Jester is sent to jail during some morning, then the game ends immediately in his sole victory. (However, the Jester does not win if he is sent to jail during some night.) Find the probability that only the Jester wins.

In how many ways can you choose $ k $ squares on a chessboard $ n \times n $ ( $ k \leq n $) so that no two of the chosen squares lie in the same row or column?

Prove that $$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a} <1$$ where $a, b$ and $c$ are positive numbers such that $abc = 1$. (G Galperin)

Sixteen people for groups of four people such that each two groups have at most two members in common. Prove that there exists a set of six people in which every group is not properly contained in it.

Prove the following identity for every $ n\in N$: $ \sum_{j\plus{}h\equal{}n,j\geq h}\frac{(\minus{}1)^h2^{j\minus{}h}\binom{j}{h}}{j}\equal{}\frac{2}{n}$

Prove that any triangle having two equal internal angle bisectors (each measured from a vertex to the opposite side) is isosceles.

Prove that $ p $ divides $ \varphi (1+a^p) , $ where $ a\ge 2 $ is a natural number, $ p $ is a prime, and $ \varphi $ is Euler's totient. [i]Cristinel Mortici[/i]

$60$ symbols, each of which is either $X$ or $O$, are written consecutively on a strip of paper. This strip must then be cut into pieces with each piece containing symbols symmetric about their centre, e.g. $O, XX, OXXXXX, XOX$, etc. (a) Prove that there is a way of cutting the strip so that there are no more than $24$ such pieces. (b) Give an example of such an arrangement of the signs for which the number of pieces cannot be less than $15$. (c) Try to improve the result of (b).

In a planar lake, every point can be reached by a straight line from the point $A$. The same holds for the point $B$. Show that this holds for every point on the segment $[AB]$, too.

Circles $\omega_1$ and $\omega_2$ touch externally at the point $O$. Points $A$ and $B$ on the circle $\omega_1$ and points $C$ and $D$ on the circle $\omega_2$ are such that $AC$ and $BD$ are common external tangents to circles. Line $AO$ intersects segment $CD$ at point $M$ and straight line $CO$ intersexts $\omega_1$ again at point $N$. Prove that the points $B$, $M$ and $N$ lie on the same straight line.

Circles $\omega$ and $\gamma$ are drawn such that $\omega$ is internally tangent to $\gamma$, the distance between their centers are $5$, and the area inside of $\gamma$ but outside of $\omega$ is $100\pi$. What is the sum of the radii of the circles? [asy] size(3cm); real lw=0.4, dr=0.3; real r1=14, r2=9; pair A=(0,0), B=(r1-r2,0); draw(A--B,dashed); draw(circle(A,r1),linewidth(lw)); draw(circle(B,r2),linewidth(lw)); filldraw(circle(A,dr)); filldraw(circle(B,dr)); label("$5$",(A+B)/2,dir(-90)); label("$\gamma$",A+r1*dir(135),dir(135)); label("$\omega$",B+r2*dir(135),dir(135)); [/asy] [i]2020 CCA Math Bonanza Individual Round #2[/i]

Find all functions $f:\mathbb{R^+}\rightarrow\mathbb{R^+}$ such that $$\frac{x+f(y)}{xf(y)}=f(\frac{1}{y}+f(\frac{1}{x}))$$ for all positive real numbers $x$ and $y$.

Let $f_1(x) = x^3+a_1x^2+b_1x+c_1 = 0$ be an equation with three positive roots $\alpha>\beta>\gamma > 0$. From the equation $f_1(x) = 0$, one constructs the equation $f_2(x) = x^3 +a_2x^2 +b_2x+c_2 = x(x+b_1)^2 -(a_1x+c_1)^2 = 0$. Continuing this process, we get equations $f_3,\cdots, f_n$. Prove that \[\lim_{n\to\infty}\sqrt[2^{n-1}]{-a_n} = \alpha\]

The line $\ell$ parallel to the side $BC$ of the triangle $ABC$, intersects its sides $AB,AC$ at the points $D,E$, respectively. The circumscribed circle of triangle $ABC$ intersects line $\ell$ at points $F$ and $G$, such that points $F,D,E,G$ lie on line $\ell$ in this order. The circumscribed circles of the triangles $FEB$ and $DGC$ intersect at points $P$ and $Q$. Prove that points $A, P$ and $Q$ are collinear.

Let $ n > 1$ be an odd positive integer and $ A = (a_{ij})_{i, j = 1..n}$ be the $ n \times n$ matrix with \[ a_{ij}= \begin{cases}2 & \text{if }i = j \\ 1 & \text{if }i-j \equiv \pm 2 \pmod n \\ 0 & \text{otherwise}\end{cases}.\] Find $ \det A$.

Given three points that are not on the same straight line. Three circles pass through each pair of the points so that the tangents to the circles at their intersection points are perpendicular to each other. Construct the circles.

Prove that the equation $$\frac{1}{\sqrt{x} +\sqrt{1006}}+\frac{1}{\sqrt{2012 -x} +\sqrt{1006}}=\frac{2}{\sqrt{x} +\sqrt{2012 -x}}$$ has $2013$ integer solutions.